1. You can always plug in your values of x, y, and z to see if they satisfy the three equations.
2. Every homogeneous system of equations has only the trivial solution (x,y,z) = (0,0,0) or infinitely many non-zero solutions in which the trivial solution is one of them (specifically when there are more unknowns than equations which isn't the case here). There are no other alternatives.
When you say you got 1 = 0, did you mean that in your augmented matrix you had a row that was similar to: . This would mean z = 0, giving your trivial solution.